On the asymptotic behavior of autonomous systems

Article type: Research Article

Authors: Sasu, Adina Luminiţa | Sasu, Bogdan^;

Affiliations: Faculty of Mathematics and Computer Science, West University of Timişoara, Timişoara, Romania E-mails: {sasu, bsasu}@math.uvt.ro

Note: [] Corresponding author: Bogdan Sasu, Faculty of Mathematics and Computer Science, West University of Timişoara, V. Pârvan Blvd. No. 4, 300223 Timişoara, Romania. E-mail: [email protected]

Abstract: The aim of this paper is to present a new and a complete study concerning the asymptotic behavior of autonomous systems in terms of input–output techniques. We characterize the exponential dichotomy of autonomous systems modeled by C0-semigroups in terms of admissibility type conditions, pointing out connections between the dichotomic splitting and the properties of the solutions of some associated control systems in certain function spaces. We deduce the structure and the uniqueness of the dichotomy projection and we obtain necessary and sufficient conditions for exponential dichotomy. Next, the main results are applied to the study of the stability and expansiveness of autonomous systems on the half-line. Throughout the paper, we present several examples which motivate the techniques and clarify the relevance of the arguments and the generality of the obtained results.

Keywords: autonomous system, dichotomy, stability, expansiveness

DOI: 10.3233/ASY-121161

Journal: Asymptotic Analysis, vol. 83, no. 4, pp. 303-329, 2013